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1 Preprint of a paper pblished in inimal Srfaces, Geometric Analysis and Symplectic Geometry Adv. Std. Pre ath, 34 (2002) SOLUTION TO THE SHADOW PROBLE IN 3-SPACE OHAAD GHOI Abstract. If a convex srface, sch as an egg shell, is illminated from any given direction, then the corresponding shadow cast on the srface forms a connected sbset. The shadow problem, first stdied by H. Wente in 1978, asks whether a converse of this phenomenon is tre as well. In this report it is shown that the answer is yes provided that each shadow is simply connected; otherwise, the answer is no. Frther, the motivations behind this problem, and some ramifications of its soltion for stdying constant mean crvatre srfaces in 3-space (soap bbbles) are discssed. 1. Introdction Let ρ R 3 be a smooth convex srface, i.e., the bondary of a convex body; let n:! S 2 denote the otward nit normal vectorfield, which we also refer to as the Gass map, of ; and let 2 S 2 be a nit vector. Sppose that is illminated by parallel rays of light flowing in the direction of, see Figre 1. Then, the shadow p n(p) Figre 1. The shadow cast on a srface, when illminated by light rays parallel to corresponds to points p in where the inner prodct between and the nit normal n(p) is positive. cast on, i.e., the set of points in not reached by the rays of light, is given by (1.1) S := f p 2 j hn(p);i>0g; 1991 athematics Sbject Classification. Primary 53A05, 53A02; Secondary 52A15, 53C45. Last Typeset September 9,

2 2 OHAAD GHOI where h ; i denotes the standard inner prodct in R 3. It is intitively clear, and not too difficlt to show [6], that if is convex, then, for every 2 S 2, S is a connected sbset of. It is natral to ask whether the connectedness of the shadows characterizes convex srfaces, i.e., whether the converse of the above phenomenon holds as well. ore precisely, let be a closed (i.e., compact and connected) srface immersed in R 3. Sppose that is oriented, so that the Gass map is globally well-defined. Then, for every nit vector 2 S 2, let the corresponding shadow, S, be defined as in (1.1). Sppose that for every, S is a connected sbset of. Does it then follow that is convex? In 1978, motivated by problems concerning the stability of constant mean crvatre srfaces, H. Wente appears to have been the first person to have stdied the above qestion [18], see Section 4, which has since become known as the shadow problem (a.k.a. the illmination conjectre). Recently, the athor has proved that this problem has a positive soltion provided that the shadows are simply connected: Theorem 1.1. Let be an oriented compact srface immersed in R 3. Sppose that for every 2 S 2, the corresponding shadow, S, is simply connected. Then is convex. In particlar, is embedded and homeomorphic to S 2. A proof of the above theorem is otlined in Section 2. Frthermore, in Section 3, we will show that the additional condition in Theorem 1.1 (the word simply) is in fact necessary, as there exist embedded closed srfaces of gens one all of whose shadows are connected. Some ramifications for stdying constant mean crvatre srfaces will be discssed in Section 4. Note 1.2. If is assmed to be simply connected, then the assmption, in Theorem 1.1, that the shadows be `simply connected' may be weakened to `connected' [6]. Note 1.3. The compactness assmption in Theorem 1.1 cannot be removed. Sppose, for instance, that is a hyperbolic paraboloid, sch as the one given by the graph of the eqation z = xy. Then all the shadows of are simply connected, even thogh is not a convex srface. To see this, let H denote the (open) hemisphere in S 2 determined by the nit vector, i.e., let H := fx 2 S 2 j hx; i > 0g. A direct comptation shows that the Gass map of, n, is a homeomorphism into H (0;0;1) (the Northern hemisphere). Frther, note that S = n 1 (H ) = n 1 (H H (0;0;1)). Ths, since H H (0;0;1) is simply connected, it follows that S is simply connected as well. 2. Otline of the Proof The proof is by contradiction, and is organized into three steps described below. The first two steps employ techniqes from orse theory [14], and the third step, which is the main part of the proof, introdces a topological invariant for shadows by permting the critical points of height fnctions. For a fll treatment of all the details, we refer the reader to [10] or [6].

3 SOLUTION TO THE SHADOW PROBLE IN 3-SPACE Critical points of height fnctions. Sppose that satisfies the hypothesis of Theorem 1.1, bt is not convex. Then there exists a nit vector v 2 S 2 sch that the corresponding height fnction h v :! R, defined by h v (p) := hp; vi; has at least three nondegenerate critical points, see Figre 2. This follows from basics p 2 p 3 v Μ p 1 Figre 2. If a closed srface,, immersed in R 3, is not convex, then there exists a direction, v, sch that the corresponding height fnction, h v, is a orse fnction with at least three critical points. of the theory of tight immersions [4] going back to the works of Chern and Lashof [5]: let #C(h ) denote the nmber of critical points of a orse height fnction h, and let K be the Gass crvatre of ; then one has the following formla 1 2 Z Z #C(h ) d = S 2 jk(x)j dx: Note that the integral on the left is well-defined, becase for almost every 2 S 2, h is a orse fnction (this is an easy application of Sard's theorem), and conseqently #C(h ) is finite. The integral on the right is known as the total absolte crvatre, which is bonded below by 4ß, becase the Gass map is srjective. R jk(x)jdx attains its minimm only when is convex [5, Thm 3]. Ths, assming that is not convex, R jk(x)jdx > 4ß. Conseqently, by the above formla, there has to exist a orse fnction, h v, with more than two critical points Reglarity of the bondary of shadows. Let v? := f 2 S 2 j h; vi = 0g. Using Sard's theorem, it can be shown that, after a pertrbation of v, we can assme that there exists a vector 0 2 v?, sch that the bondary of the corresponding 0, is a reglar sbmanifold of. This is a conseqence of the fact that, for almost every 2 S is reglar, which, briefly, may be proved as follows: define the shadow fnction f :! R by f (p) := hn(p);i; and observe ρ f 1 (0). Frther, let UT denote the nit tangent bndle of, i.e., UT := f(p; t p ) j p 2 ; t p 2 T p ; and kt p k = 1g. Define fi : UT! S 2

4 4 OHAAD GHOI and ß : UT!, by fi(p; t p ) := t p and ß(p; t p ) := p respectively. UT? y ß! fi S 2 Then f 1 (0) = ß(fi 1 ()). Let be a reglar vale of fi. Then fi 1 () is a reglar crve in UT. Frther, it is not too difficlt to show that ß is an embedding on fi 1 (). Hence, by Sard's theorem, for almost every, f 1 (0), and is a reglar crve (for more reslts of this type and an introdction to stdying geometry of the shadow bondaries on illminated srfaces see [11]). After a rotation of the coordinate axis, and for the sake of convenience, we assme from now on that v = (0; 0; 1) and 0 = (1; 0; 0). Frther, we parameterize v? by ( ) := (cos ; sin ; 0), 2 [0; 2ß] Indced permtations on the critical points. Let p i, i = 1; 2; 3, denote three critical points of h v. For every 2 [0; 2ß], we define a permtation ff( ) 2 Sym(p 1 ;p 2 ;p 3 ), the symmetric grop of three elements, as follows. Fix 2 [0; 2ß]. Note that p i ( ); frthermore, since p i is a nondegenerate critical point of the height fnction h v, it follows ( ) is reglar in a neighborhood of p i, see Figre 3. This together with the simply-connectedness of S ( ) n(p) p p S S 2 S Figre 3. If p is a reglar critical point of the height fnction h v, then n(p) = ±v, and for every 2 S 2 orthogonal to n(p) the bondary of the corresponding is reglar in a neighborhood of p; becase, n is a local diffeomorphism at p, is the pll-back via n of a great circle in S 2. implies that there exists a simple closed crve T in the closre of S ( ) sch that: (i) T is composed of three smooth arcs which end at p i, (ii) each arc ( ) transversally, and (iii) the interior of each arc lies in S ( ). We say that sch a crve is a standard triangle for S ( ), see Figre 4. Since S ( ) is simply connected, T bonds a niqe region in S ( ). This region inherits an orientation from (recall that is, by assmption, oriented), which in trn indces a preferred sense of direction on T. The indced direction on T determines a permtations for p i in a natral way; for instance, sppose that as we move along T away from p 1 we enconter p 2

5 SOLUTION TO THE SHADOW PROBLE IN 3-SPACE 5 p 1 Τ p 2 p 3 S ( ) θ Μ Figre 4. Each shadow, S ( ), contains a standard triangle. Note that the bondary of the shadow is a reglar crve in a neighborhood of the critical points p i. before reaching p 3, then we say that the indced permtation is the cycle (p 1 p 2 p 3 ). Finally, note that the indced permtation on p i does not depend on the choice of the standard triangle; becase, if T 0 is any other standard triangle in S ( ), then T 0 and T are homotopic in S ( ) by the simply-connectedness of S ( ). So we conclde that each shadow S ( ) determines a niqe permtation on fp 1 ;p 2 ;p 3 g, which we denote by ff( ). We claim that the map ff : [0; 2ß]! Sym(p 1 ;p 2 ;p 3 ) which we defined above is constant. To this end, since [0; 2ß] is connected, it sffices to show that ff is locally constant. This follows from the fact that whenever and 0 are sfficiently close, then S ( ) and S ( 0 ) have a standard triangle in common, see Figre 5. The proof p 1 Τ p 2 p 3 S ( θ + ε ) Μ Figre 5. For every there exists an ffl > 0 sch that the shadows S ( ) and S ( +ffl) have a standard triangle in common. This shows that the indced permtation on fp 1 ;p 2 ;p 3 g is locally constant. of this is based on the compactness of T, the assmption that T ( ) only at p i and does so transversally, and the observation that in a neighborhood of p ( ) depends continosly on.

6 6 OHAAD GHOI On the other hand, it is not difficlt to show that ff(0) 6= ff(ß). To see this, recall (0) is reglar by constrction. This implies (ß) (0). Frther, since S (0) is, by assmption, simply (0) is connected. In (0) is a simple closed crve passing throgh p i. Sppose (0) is given the orientation indced by S (0), and note that the corresponding permtation indced on p i coincides with ff(0), becase all standard triangles in S (0) are homotopic (0). Similarly, ( ) is oriented by S (ß), then this gives rise to a permtation of p i which is identical with ff(ß). S (0) and S (ß) indce opposite orientations ( ). Hence ff(0) = ff(ß), which prodces the desired contradiction and completes the proof. 3. A Conterexample In this section we show that Theorem 1.1 does not remain valid if the condition that the shadows be `simply connected' is replaced by `connected'. ore specifically, we show that there exists a smooth embedded srface of gens one all of whose shadows are connected. This srface is given by bilding a tbe arond a closed crve withot any pairs of parallel tangent lines. An explicit example of sch a crve, formlated by Ralph Howard, is given by fl(t) := (x(t);y(t);z(t)), where (3.1) x(t) := cos(t) 1 20 cos(4t) cos(2t); y(t) := + sin(t) sin(2t) sin(4t); z(t) := sin(3t) 2 cos(3t) sin(3t); 15 t 2 [0; 2ß]. Figre 6 shows the pictres of a small tbe bilt arond the above crve. Let denote the trace of fl. Since is a reglar sbmanifold, it follows from the Figre 6. Three different views of a nonconvex srface all of whose shadows are connected. This srface is constrcted by bilding a tbe arond a crve with no pair of parallel tangents.

7 SOLUTION TO THE SHADOW PROBLE IN 3-SPACE 7 tblar neighborhood theorem that there exists an r > 0 sch that := fx 2 R 3 j dist(x; ) = rg is a smooth srface, where dist(x; ) := inf y2 kx yk. We claim that, since has no pair of parallel tangent lines, each shadow of is a connected sbset. Before proving this, however, we describe a general procedre for constrcting. Let T ρ S 2 be a smooth simple closed crve sch that (i) the origin is contained in the interior of the convex hll of T, (0; 0; 0) 2 int conv T, and (ii) T does not contain any pair of antipodal points. Althogh it is not immediately clear that sch crves exist, they are not difficlt to constrct. Figre 7 shows an example, which is perhaps, qalitatively speaking, the simplest. Let T (s), s 2 R, denote a periodic Figre 7. A simple closed crve on the sphere which contains the origin in the interior of its convex and is disjoint from its antipodal reflection. An appropriate integration of the above yields a space crve with no parallel tangents. parameterization of T by arclength. So, assming T has total length L, we have T (s + L) = T (s). Since (0; 0; 0) 2 int conv T, there exists a (density) fnction v(s) with period L sch that R L 0 v(s)t (s) ds = 0; or, intitively speaking, it is possible to distribte mass along T so that the center of gravity of the reslting object coincides with the origin. Now set fl(t) := Z t 0 v(s)t (s) ds: Then fl(t + L) = fl(t). Frther, fl 0 (t)=kfl 0 (t)k = T (t). Ths fl is a closed crve whose tangential spherical image coincides with T. In particlar, fl has no parallel tangent lines. Hence (the trace of fl) is the desired crve. Next we show that, given by a small tbe arond, has connected shadows. To see this, let ß :! be the obvios projection, i.e., the nearest point map. For every x 2, let F x := ß 1 (x) be the corresponding fiber. Note that (i) each fiber, F x, is a circle, (ii) the image of each fiber nder the Gass map, n(f x ), is the great circle in S 2 which lies in the plane perpendiclar to T (x), and (iii) n is one-to-one on each F x. Let 2 S 2, and let S be the corresponding shadow cast on. Recall that S = n 1 (H ), where H := fx 2 S 2 j hx; i > 0g is an open hemisphere, see Figre 8. Ths, for each fiber, F x, we have only two possibilities:

8 8 OHAAD GHOI F x n( F x ) x T(x) T(x) Γ Figre 8. Unless T (x) and are parallel, the fiber F x of the tbe arond intersects the shadow S along an open semicircle. either F x intersects S in an open half-circle, or F x is disjoint from S. Bt, by constrction of, the latter occrs for at most for one x 2. Hence, it follows that each shadow, S, is either homeomorphic to a disk or an annls. In particlar, S is connected for every 2 S 2. Note 3.1. It is an elementary and well-known fact that if is a closed crve, then its tangential spherical image (a.k.a. tangent indicatrix or tantrix), contains the origin in the interior of its convex hll. Here we showed that a converse of this phenomenon holds as well. This observation is also known, and has been attribted to Löwner; bt it is not clear if it had ever been pblished by him. See [12] for detailed proofs and historical comments. A proof may also be fond in [8, p. 168] Note 3.2. It is possible to constrct a simple closed crve withot parallel tangents which lies on a cylinder with a convex base. In fact, the eqations (3.1) give one sch example. So a loop withot parallel tangents may lie on the bondary of a convex body. Interestingly enogh, however, no sch crve may be constrcted on an ellipsoid. This follows from recent reslts of Joel Weiner [21] or Brce Solomon [17] who showed that the tantrix of a spherical crve, if embedded, divides the sphere into eqal areas. Conseqently, any loop on a sphere has to have a pair of parallel tangents. Frther, ellipsoids mst have this property as well, becase they are eqivalent to the sphere p to a linear transformation. It wold be interesting to know if ellipsoids are the only closed srfaces which admit no loops withot parallel tangent Applications 4.1. Stable Constant ean Crvatre Srfaces. In this section we discss the original motivation for stdying the shadow problem, and indicate how one can obtain a classical isoperimetric reslt sing Theorem 1.1. Let be an oriented, closed, and stable constant mean crvatre (CC) srface immersed in R 3. Stable means that is a critical srface for the area fnctional 1 Note added in proof: since this paper was first written, the athor and Brce Solomon have proved that the property of having no loops withot parallel tangent lines (skew loops), does indeed characterize ellipsoids amongst all closed srfaces immersed in 3-space [7].

9 SOLUTION TO THE SHADOW PROBLE IN 3-SPACE 9 sbject to a volme constraint. In 1978, when the shadow problem seems to have first originated, it was not yet known that is necessarily a (rond) sphere. otivated by this qestion, one might make the following observation:, mch like a sphere, has connected shadows. This is based on a variation argment, described below, which the athor first learned from Henry Wente [18]. For all 2 S 2, the shadow fnction f :! R, f (p) := hn(p);i, is a Jacobi field on, i.e., for the pertrbation p 7! p + tf (p)n(p); the first variation of volme and the first and second variation of area are all zero; becase, the variations corresponds to a rigid motion of in the direction. Consider the nodal regions of f on. These are the sets where f is either positive or negative, and correspond, therefore, to the shadows S and S, respectively. Sppose, towards a contradiction, that S is not connected, then there are at least three nodal regions A i, i = 1; 2; 3. Conseqently, one can form three fnctions f i by setting f i := f on A i and f i := 0 elsewhere. One can then take a sitable linear combination P 3 i=1 if i, to obtain a fnction for which the first variation of volme is zero bt the second variation of area is negative, contradicting the stability assmption. Hence, we conclde that all shadows of are connected. Sppose now that is simply connected, then, see Note 1.2, the connectedness of the shadows of imply that each shadow is simply connected. Hence, by Theorem 1.1, it follows that is convex. In particlar, is embedded. Conseqently, by applying the maximm principle together with the reflection techniqe introdced by Aleksandrov [1], it follows that is a sphere. The above reslt is well-known, and may be regarded as a weak version of a theorem of Hopf [9, p. 138], or a theorem of Barbosa and do Carmo [2]. Hopf showed, withot assming stability, that any closed CC srface of gens zero mst be a sphere, and Barbosa and do Carmo proved that a closed oriented srface of higher gens mst also be a sphere provided it is stable (for an elementary proof of this reslt, see [19]). Finally, Wente showed that the stability assmption in higher gens is not sperflos [20] by constrcting a CC tors in R 3 ; ths, settling a famos and long standing qestion of Hopf [9, Pg. 131]. In closing this section, we shold also point ot that a nmber of reslts concerning the connection between the nmber of components of nodal regions of the shadow fnction (the vision nmber) and the stability index of complete minimal srfaces in R 3 have been obtained by Jaigyong Choe [3] Convexity of the level sets of H-graphs. Recently, shadows on illminated srfaces have been stdied within the context of another problem involving constant mean crvatre. This problem, nlike those mentioned in the previos sbsection, is still open. Let Ω ρ R 2 be a convex domain, and f 2 C 2 (Ω) C 0 (Ω) be a soltion to the following bondary vale problem: grad f Div( p ) = 2H on Ω; and f = k grad fk 2

10 10 OHAAD GHOI Let denote the graph of f. Then has constant mean crvatre H. Intitively, one may think of as the membrane of least area, spanned which traps a given volme above the xy-plane. It has been a well-known and long standing problem [13] to show that the level sets of, and those given by eqations of similar type, are convex. Recently, John ccan [15] has obtained a nmber of reslts on this problem. In particlar, he has shown that for every nit vector ( ) := (cos ; sin ; 0), the set X ( ) := fx 2 Ω j hgrad f(x);( )i = 0g is a connected reglar crve, assming has strictly positive crvatre. This implies that the shadow S ( ) is a simply connected sbset of, becase X ( ) is the projection ( ) into the xy-plane. One is then led to consider the following qestion [16]: does the simply-connectedness of the shadows S ( ) imply that the level sets of are convex? The answer is negative, see Figre 9. At present it is not clear what Figre 9. A graph with zero bondary vales over a convex domain which has nonconvex level sets, even thogh the shadows S ( ) are simply connected. S ( ) is simply connected becase the graph has a niqe critical point. shadow property, if any, wold characterize the convexity of level sets. Acknowledgments The athor is indebted to Ralph Howard for a good deal of help and advice throghot this project. Frther, the athor wishes to thank John ccan for bringing the shadow problem to the athor's attention, Henry Wente for his informative comments and encoragement, and Brce Solomon for his interest in aspects of this work. References [1] A. D. Aleksandrov,Uniqeness theorems for srfaces in the large. I, Amer. ath. Soc. Transl, (2) 21 (1962), [2] J. L. Barbosa &. do Carmo, Stability of hypersrfaces with constant mean crvatre, ath. Z. 185 (1984), no. 3, [3] J. Choe, Index, vision nmber and stability of complete minimal srfaces, Arch. Rational ech. Anal. 109 (1990), no. 3, [4] T. Cecil, & S. S. Chern, Tight and tat sbmanifolds, Cambridge University Press, Cambridge, 1997.

11 SOLUTION TO THE SHADOW PROBLE IN 3-SPACE 11 [5] S. S. Chern, & R. K. Lashof, On the total crvatre of immersed manifolds, Amer. J. ath. 79 (1957), ; On the total crvatre of immersed manifolds. II, ichigan ath. J. 5 (1958), [6]. Ghomi, Shadows and convexity of srfaces, preprint available at [7]. Ghomi, and B. Solomon, Skew loops and qadric srfaces, preprint available at [8]. Gromov, Partial differential relations, Ergebnisse der athematik nd ihrer Grenzgebiete (3) [Reslts in athematics and Related Areas (3)], 9. Springer-Verlag, Berlin-New York, [9] H. Hopf, Differential geometry in the large, Springer-Verlag, Berlin-New York, [10] R. Howard, ohammad Ghomi's Soltion to the Shadow Problem, Lectre Notes, available at [11] The Geometry of Shadow Bondaries on Srfaces in Space, Lectre Notes, available at [12] Characterization of Tantrix Crves, Lectre Notes, available at [13] B. Kawohl, Rearrangements and convexity of level sets in PDE, Springer-Verlag, Berlin-New York, [14] J. ilnor, orse theory, Princeton University Press, Princeton, N.J.,1963. [15] J. ccan, Contina of H-graphs: convexity and isoperimetric stability, Calc. Var. Partial Differential Eqations 9 (1999), no. 4, [16] Personal , Jne 23, [17] B. Solomon, Tantrices of spherical crves, Amer. ath. onthly 103 (1996), no. 1, [18] H. C. Wente, Personal , Janary 9, [19] A note on the stability theorem of J. L. Barbosa and. Do Carmo for closed srfaces of constant mean crvatre, Pacific J. ath. 147 (1991), no. 2, [20] Conterexample to a conjectre of H. Hopf, Pacific J. ath. 121 (1986), no. 1, [21] J. Weiner, Flat tori in S 3 and their Gass maps, Proc. London ath. Soc. (3) 62 (1991), no. 1, Department of athematics, University of Soth Carolina, Colmbia, SC address: ghomi@math.sc.ed URL: